Differential Geometry : 18.950 (Spring 2006)

Lectures

Textbooks

Graders

Examinations and Homework

• HW #1: Due Thursday, March 2. Ch 2.2: 5, 12, 13. Ch 2.3: 1, 3, 4. Ch 2.4: 1, 3, 12, 13, 17, 18, 21, 24. Ch. 2.5: 1, 3, 14.
• HW #2: Due Tuesday March 21. Ch 3.2: 8. Ch 3.3: 1, 2, 5, 6, 7, 16, 20, 21, 22.
• HW #3: Due Thursday April 13. Ch 3.5: 12, 13, 14.
• HW #4: Due Tuesday May 9.

Brief Lecture notes

• Lecture 1: Tuesday, February 7
  
  1. Introduction and Outline.
  2. Curves in R^n.
  3. Curvature and torsion of curves in R^3.
  4. Fundamental theorem of the local theory of curves.
• Lecture 2: Thursday, February 9
  
  1. Isoperimetric Inequality
  2. 2-dimensional surfaces in R^3.
• Lecture 3: Tuesday, February 14
  
  1. Stereographic projection.
  2. More on regular surfaces.
  3. Overlap maps are diffeomorphisms.
  4. Critical and regular values.
• Lecture 4: Thursday, February 16
  
  1. Surfaces of revolution.
  2. Tangent plane
  3. Differential of maps between surfaces.
  4. First Fundamental Form.
  5. Surfaces of revolution
• Holiday, no class on Tuesday, February 21 (Don't forget to go to your Monday classes instead!)
  
• Lecture 5: Thursday, February 23
  
  1. Torus.
  2. Orientation.
  3. Orientable and non-orientable surfaces.
• Lecture 6: February 28
  
  1. Area of surfaces.
  2. Transformation rule for vector, 1-forms, and tensors.
  3. The Gauss Map, differential of Gauss Map.
• Lecture 7: March 2
  
  1. More on Gauss Map and its differential.
  2. Self-Adjoint Linear Maps.
  3. Normal curvature.
• Lecture 8: March 7
  
  1. Diagonalize quadratic forms.
  2. Diagonalize symmetric matrix by orthogonal matrices.
  3. Computations of normal curvature using normal sections.
• Lecture 9: March 9
  
  1. Gaussian curvature and mean curvature.
  2. Elliptic, hyperbolic, parabolic and planar points.
  3. Computations for a hyperbolic paraboloid.
  4. Umbilic points.
• Lecture 10: March 14
  
  1. General formula for second fundamental form, Gaussian curvature, mean curvature in coordinates.
  2. Gaussian curvature of a torus.
  3. Differential equation for asymptotic curves.
• Lecture 11: March 16
  
  1. Differential equation for lines of curvature.
  2. Gaussian and mean curvature for surfaces of revolution and graphs.
• Lecture 12: Tuesday, March 21
  
  1. Connected and umbilic -> contained in sphere or plane.
  2. Relation between Gaussian curvature and area of Gauss map image.
  3. Review for Exam.
• Thursday, March 23
  
  1. Exam I.
• Lecture 13: April 4
  
  1. Variation of area functional.
  2. Minimal surfaces.
  3. The catenoid.
• Lecture 14: April 6
  
  1. The helicoid.
  2. Isometries and local isometries.
  3. Catenoid and Helicoid are locally isometric.
• Lecture 15: April 11
  
  1. Locally conformal maps.
  2. The map J : T_p S -> T_p S.
  3. Cauchy-Riemann equations.
• Lecture 16: April 13
  
  1. More on Cauchy-Riemann equations.
  2. The gradient.
  3. The Laplacian
  4. Existence of isothermal coordinates.
• Patriots Day, no class on April 18
  
• Lecture 17: April 20
  
  1. Tensor product of vector spaces.
  2. Exterior product of vector spaces.
  3. Wedge product.
• Lecture 18: April 25
  
  1. Dual spaces.
  2. Differential p-forms on surfaces.
  3. Exterior differential d.
• Lecture 19: April 27
  
  1. Integral of 2-forms on oriented surfaces.
  2. Surfaces with boundary.
  3. Stokes' Theorem for triangles.
  4. Hodge star-operator.
• Lecture 20: May 2
  
  1. Stokes' Theorem for compact oriented surfaces.
  2. Review of inner product on \Lambda^p(T^*S).
  3. More on Hodge star-operator.
  4. Divergence operator
  5. Laplacian on forms.
  6. Integration by parts formula on surfaces.
• Lecture 21: May 4
  
  1. Cone and plane are locally isometric.
  2. Gaussian curvature of a cone.
  3. Christoffel symbols.
  4. Gauss' Theorema Egregium.
• Lecture 22: May 9
  
  1. Parallel Transport
  2. Geodesics
• May 11
  
  1. Exam II.
• Lecture 23: May 16
  
  1. The Gauss-Bonnet Theorem (local version)
• Lecture 24: May 18
  
  1. Global version of Gauss-Bonnet Theorem.
  2. Applications of Gauss-Bonnet.